The methods of analysis of primary signals considered above make it possible to determine their spectral and energy characteristics. Primary signals are the main carriers of information. At the same time, their spectral characteristics do not correspond to the frequency characteristics of transmission channels of radio technical information systems. As a rule, the energy of the primary signals is concentrated in the low frequency region. So, for example, when transmitting speech or music, the energy of the primary signal is concentrated approximately in the frequency range from 20 Hz to 15 kHz. At the same time, the decimeter wavelength range, which is widely used for the transmission of information and music programs, occupies frequencies from 300 to 3000 megahertz. The problem arises of transferring the spectra of the primary signals to the corresponding radio frequency ranges for their transmission over radio channels. This task is accomplished by means of a modulation operation.

Modulation is the procedure for converting low-frequency primary signals into radio frequency signals..

The modulation procedure involves the primary signal and some auxiliary oscillation, called bearing vibration or just a carrier. In general, the modulation procedure can be represented as follows

where is the transformation rule (operator) of the primary signal into a modulated oscillation.

This rule indicates which parameter (or several parameters) of the carrier oscillation changes according to the law of change. Since it controls the change of parameters, then, as noted in the first section, the signal is a control (modulating) signal, but a modulated signal. Obviously, it corresponds to the operator of the generalized structural diagram of RTIS.

Expression (4.1) allows us to classify the types of modulation, which is shown in Fig. 4.1.

Rice. 4.1

As the classification features, we choose the type (form) of the control signal, the form of the carrier wave and the type of the controlled parameter of the carrier wave.

In the first section, the classification of primary signals was carried out. In radio engineering information systems, continuous and digital signals are most widely used as primary (control) signals. In accordance with this, by the type of the control signal, one can distinguish continuous and discrete modulation.

In practical radio engineering, harmonic oscillations and pulse sequences are used as carrier oscillations. In accordance with the form of the carrier vibration, they are distinguished harmonic carrier modulation and pulse modulation.

And finally, according to the form of the controlled parameter of the carrier oscillation in the case of a harmonic carrier, one can distinguish amplitude, frequency and phase modulation... Obviously, in this case, the amplitude, frequency, or the initial phase of the harmonic oscillation act as a controlled parameter, respectively. If a pulse sequence is used as a carrier wave, then the analogue of frequency modulation is pulse width modulation, where the controlled parameter is the pulse duration, and the analogue of phase modulation is time pulse modulation, where the controlled parameter is the position of the pulse on the time axis.

In modern radio engineering systems, the harmonic wave is most widely used as the carrier wave. Considering this circumstance, in the future, the main attention will be paid to signals with continuous and discrete modulation of a harmonic carrier.

4.2. Continuous amplitude modulation signals

Let us begin the consideration of modulated signals with signals in which the variable parameter is amplitude carrier vibration. The modulated signal in this case is amplitude modulated or amplitude modulated signal (AM signal).

As noted above, the main attention will be paid to signals, the carrier vibration of which is a harmonic vibration of the form

where is the amplitude of the carrier vibration,

Is the frequency of the carrier wave.

Consider continuous waveforms first as baseband waveforms. Then the modulated signals will be signals with continuous amplitude modulation... Such a signal is described by the expression

where is the envelope of the AM signal,

Is the amplitude modulation factor.

From expression (4.2) it follows that the AM signal is the product of the envelope and the harmonic function. Amplitude modulation factor characterizes modulation depth and is generally described by the expression

. (4.3)

Obviously, when the signal is just a carrier wave.

For a more detailed analysis of the characteristics of AM signals, let us consider the simplest AM signal, in which a harmonic oscillation acts as a modulating signal

, (4.4)

where, - respectively the amplitude and frequency of the modulating (control) signal, and. In this case, the signal is described by the expression

, (4.5)

and is called a single tone amplitude modulation signal.

In fig. 4.2 shows the baseband waveform, the carrier waveform and the waveform.

For such a signal, the amplitude modulation depth coefficient is

Using the well-known trigonometric relation

after simple transformations we get

Expression (4.6) sets the spectral composition of a single-tone AM signal. The first term is an unmodulated waveform (carrier waveform). The second and third terms correspond to new harmonic components, which appeared as a result of modulation of the amplitude of the carrier oscillation; the frequency of these vibrations and are called the lower and upper side frequencies, and the components themselves are called the lower and upper side frequencies.

The amplitudes of these two oscillations are the same and amount to

, (4.7)

In fig. 4.3 shows the amplitude spectrum of a single-tone AM signal. It follows from this figure that the amplitudes of the lateral components are located symmetrically with respect to the amplitude and the initial phase of the carrier wave. Obviously, the spectrum width of a single-tone AM signal is equal to twice the frequency of the control signal

In the general case, when the control signal is characterized by an arbitrary spectrum concentrated in the frequency band from to, the spectral nature of the AM signal does not fundamentally differ from the single-tone one.

In fig. 4.4 shows the spectra of the control signal and the signal with amplitude modulation. Unlike a single tone AM signal, the spectrum of an arbitrary AM signal includes lower and upper sidebands. In this case, the upper sideband is a copy of the control signal spectrum, shifted along the frequency axis by

value, and the lower sidebar is a zekal display of the upper one. Obviously, the spectrum width of an arbitrary AM signal is

those. equal to twice the upper cutoff frequency of the control signal.

Let's return to the single-tone amplitude modulation signal and find its energy characteristics. The average power of the AM signal over the period of the control signal is determined by the formula:

. (4.9)

Since, a, we put , where . Substituting expression (4.6) into (4.9), after simple but rather cumbersome transformations, taking into account the fact that and using trigonometric relations

Here, the first term characterizes the average power of the carrier vibration, and the second - the total average power of the lateral components, i.e.

Since the total average power of the lateral components is divided equally between the lower and upper, which follows from (4.7), it follows from this

Thus, more than half of the power (taking into account that) is spent on the transmission of the carrier wave in the AM signal than on the transmission of side components. Since the information is embedded precisely in the lateral components, the transmission of the component of the carrier vibration is impractical from an energy point of view. The search for more effective methods of using the principle of amplitude modulation leads to signals of balanced and single-sideband amplitude modulation.

4.3. Balanced and SSB signals

Balanced amplitude modulation (BAM) signals are characterized by the absence of a carrier vibration component in the spectrum. Let us proceed directly to the consideration of signals of single-tone balanced modulation, when a harmonic signal of the form (4.4) acts as a control oscillation. Exclusion from (4.6) of the component of the carrier vibration

leads to the result

Let's calculate the average power of the balanced modulation signal. Substitution of (4.12) into (4.9) after transformations gives the expression

.

Obviously, the energy gain when using balanced modulation signals in comparison with classical amplitude modulation will be equal to

In this case, the gain is.

In fig. 4.5 shows one of the variants of the block diagram of the balanced amplitude modulation signal generator. The shaper contains:

• Inv1, Inv2 - signal inverters (devices that change the polarity of voltages to the opposite);
• AM1, AM2 - amplitude modulators;
• SM - adder.

The oscillation of the carrier frequency is fed directly to the inputs of the modulators AM1 and AM2. As for the control signal, it goes directly to the second input AM1, and to the second input AM2 through the inverter Inv1. As a result, oscillations of the form are formed at the outputs of the modulators

The inputs of the adder receive, respectively, oscillations and ... The resulting signal at the output of the adder will be

In the case of single-tone amplitude modulation, expression (4.13) takes the form

Using the formula for the product of cosines, after transformations we obtain

which, up to a constant factor, coincides with (4.12). Obviously, the width of the spectrum of the BAM signals is equal to the width of the spectrum of the AM signals.

Balanced amplitude modulation eliminates carrier wave transmission, resulting in energy gain. However, both sidebands (sidebands in the case of a single tone AM) carry the same information. This suggests a conclusion about the advisability of generating and transmitting signals with suppressed one of the side bands. In this case, we come to single sideband amplitude modulation (OAM).

If one of the lateral components (say, the upper lateral component) is excluded from the spectrum of the BAM signal, then in the case of a harmonic control signal, we obtain

Since the average power of the BAM signal is divided equally between the lateral components, it is obvious that the average power of the OAM signal will be

The energy gain in comparison with amplitude modulation will be

and at it will be equal.

The formation of a single-sideband AM signal can be carried out on the basis of balanced modulation signal conditioners. The block diagram of a single-sideband AM signal generator is shown in Fig. 4.6.

The SSB signal conditioner includes:

The following signals are received at the BAM1 inputs:

Then, at its output, in accordance with (4.15), a signal is generated

Signals are received at the inputs of BAM2

and .

From the output of BAM2, the oscillation described in accordance with (4.14) is removed with the replacement of cosines by sines

Taking into account the well-known trigonometric relation

the output signal of BAM2 is converted to the form

The addition of signals (4.17) and (4.18) in the adder SM gives

which, up to a constant factor, coincides with (4.16). As for the spectral characteristics, the width of the spectrum of OAM signals is half that of the spectrum of AM or BAM signals.

Thus, with the same and single-sideband AM provides a significant energy gain in comparison with classical AM and balanced modulation. At the same time, the implementation of balanced amplitude and single-sideband amplitude modulation signals is associated with some difficulties related to the need to restore the carrier wave when processing signals on the receiving side. This problem is solved by the synchronization devices of the transmitting and receiving sides, which in general leads to the complication of the equipment.

4.4. Continuous Angle Modulated Signals

4.4.1. Generalization of Angle Modulated Signals

In the previous section, the modulation procedure was considered, when the information parameter changed in accordance with the law of the control (modulating) signal was the amplitude of the carrier wave. However, in addition to the amplitude, the carrier vibration is also characterized by the frequency and the initial phase

where is the total phase of the carrier wave, which determines the current value of the phase angle.

Change either or according to the control signal corresponds to angular modulation... Thus, the concept of angular modulation includes both frequency(World Cup) and phase(FM) modulation.

Let's consider the generalized analytical relationships for signals with angle modulation. At frequency modulation in accordance with the control signal, the instantaneous frequency of the carrier oscillation changes in the range from the lower to the cutoff frequencies

The largest value of the frequency deviation from is called deviation frequency

.

If the cutoff frequencies are located symmetrically with respect to, then the frequency deviation

. (4.22)

It is this case of frequency modulation that will be considered below.

The total phase change law is defined as the integral of the instantaneous frequency. Then, taking into account (4.21) and (4.22), we can write

Substituting (4.23) into (4.20), we obtain a generalized analytical expression for the signal with frequency modulation

Term is the frequency modulation component of the total phase. It is easy to make sure that full phase frequency modulated signal changes by the integral law from.

At phase modulation, in accordance with the modulating signal, the initial phase of the carrier oscillation changes in the range from the lower to the upper limit values ​​of the phase

The largest deviation of the phase shift from is called the phase deviation. If and are located symmetrically with respect to, then ... In this case, the total phase of the phase modulated signal is

Then, substituting (4.26) into (4.20), we obtain a generalized analytical expression for the signal with phase modulation

Let us consider how the instantaneous frequency of the signal changes during phase modulation. It is known that the instantaneous frequency and current field are

phase are related by the relation

.

Substituting formula (4.26) into this expression and performing the differentiation operation, we obtain

where - the frequency component due to the presence of phase modulation of the carrier oscillation (4.20).

Thus, a change in the initial phase of the carrier wave leads to a change in the instantaneous frequency values ​​according to the time derivative law.

Practical implementation of devices for generating signals of angular modulation can be carried out by one of two methods: direct or indirect. In the direct method, in accordance with the law of change in the control signal, the parameters of the oscillatory circuit of the carrier oscillator are changed. The output signal is then frequency modulated. To obtain a phase modulation signal, a differentiating circuit is switched on at the input of the frequency modulator.

Phase modulation signals with the direct method are formed by changing the parameters of the oscillatory circuit of the amplifier connected to the output of the carrier oscillator. To convert the phase modulation signals into a frequency modulation signal, the control oscillation is applied to the input of the phase modulator via an integrating circuit.

Indirect methods do not imply a direct influence of the control signal on the parameters of the oscillatory circuit. One of the indirect methods is based on the conversion of amplitude modulated signals into phase modulation signals, and those, in turn, into frequency modulation signals. In more detail, the issues of the formation of signals of frequency and phase modulation will be discussed below.

4.4.2. Frequency modulated signals

We begin our analysis of the characteristics of angularly modulated signals by looking at single-tone frequency modulation. The control signal in this case is an oscillation of a unit amplitude (it can always be reduced to this form)

, (4.29)

and the modulated parameter of the carrier wave is the instantaneous frequency. Then, substituting (4.29) into (4.24), we obtain:

After performing the integration operation, we arrive at the following expression for the single-tone frequency modulation signal

Attitude

called index frequency modulation and has the physical meaning of the part of the frequency deviation per unit frequency of the modulating signal. So for example, if the deviation of the carrier frequency in MHz is , and the frequency of the control signal is kHz, then the frequency modulation index will be. In expression (4.30), the initial phase is not taken into account as having no fundamental significance.

The signal timing diagram for single-tone FM is shown in Fig. 4.7

We will start considering the spectral characteristics of the FM signal with a special case small frequency modulation index. Using the ratio

we represent (4.30) in the form

Since, then you can use the approximate representations

and expression (4.31) takes the form

Using the well-known trigonometric relation

and putting and, we get:

This expression resembles expression (4.6) for a single-tone AM signal. The difference is that if in a single-tone AM signal the initial phases of the side components are are the same, then in a single-tone FM signal at low frequency modulation indices they differ by an angle, i.e. are in antiphase.

The spectral diagram of such a signal is shown in Fig. 4.8

The values ​​of the initial phase of the lateral components are indicated in brackets. Obviously, the width of the spectrum of the FM signal at small indices of frequency modulation is equal to

.

Signals with low frequency modulation are rarely used in practical radio engineering.

In real radio engineering systems, the frequency modulation index significantly exceeds unity.

For example, in modern analog mobile communication systems that use frequency modulation signals for the transmission of voice messages at the upper frequency of the speech signal in kHz and frequency deviation kHz, the index, as it is easy to see, reaches a value of ~ 3-4. In VHF broadcasting systems, the frequency modulation index can exceed a value equal to 10. Therefore, let us consider the spectral characteristics of FM signals at arbitrary values ​​of the value.

Let's return to expression (4.32). The following types of decomposition are known

where is the Bessel function of the first kind of the th order.

Substituting these expressions in (4.32), after simple but rather cumbersome transformations using the already repeatedly mentioned relations of the products of cosines and sines, we obtain

(4.36)

where .

The resulting expression is the decomposition of a single-tone FM signal into harmonic components, i.e. amplitude spectrum. The first term of this expression is the spectral component of the carrier frequency oscillation with an amplitude ... The first sum of expression (4.35) characterizes the side components with amplitudes and frequencies, i.e. the lower sideband, and the second sum is the sidebands with amplitudes and frequencies, i.e. the upper side band of the spectrum.

The spectral diagram of the FM signal for an arbitrary one is shown in Fig. 4.9.

Let us analyze the nature of the amplitude spectrum of the FM signal. First of all, we note that the spectrum is symmetric with respect to the frequency of the carrier vibration and is theoretically infinite.

The components of the side side stripes are located at a distance Ω from each other, and their amplitudes depend on the frequency modulation index. And finally, for the spectral components of the lower and upper side frequencies with even indices, the initial phases coincide, and for the spectral components with odd indices they differ by an angle.

Table 4.1 shows the values ​​of the Bessel function for various i and . Let's pay attention to the component of the carrier vibration. The amplitude of this component is ... From table 4.1 it follows that at the amplitude, i.e. the spectral component of the carrier vibration in the spectrum of the FM signal is absent. But this does not mean that there is no carrier oscillation in the FM signal (4.30). The energy of the carrier vibration is simply redistributed between the components of the side bands.

Table 4.1

As already emphasized above, the FM spectrum - the signal is theoretically infinite. In practice, the bandwidth of radio devices is always limited. Let us estimate the practical width of the spectrum at which the reproduction of the FM signal can be considered undistorted.

The average power of the FM signal is defined as the sum of the average powers of the spectral components

The calculations showed that about 99% of the energy of the FM signal is concentrated in frequency components with numbers. This means that the frequency components with numbers can be neglected. Then the practical width of the spectrum for a single-tone FM, taking into account its symmetry with respect to

and for large values

Those. equal to twice the frequency deviation.

Thus, the width of the spectrum of the FM signal is approximately in times greater than the width of the spectrum of the AM signal. At the same time, for the transmission of information, it is used all energy signal. This is the advantage of FM signals over AM signals.

4.5. Discrete Modulated Signals

The above signals with continuous modulation are mainly used in radio broadcasting, radiotelephony, television and others. At the same time, the transition to digital technologies in radio engineering, including in the aforementioned areas, has led to the widespread use of signals with discrete modulation or manipulation. Since historically, discrete modulation signals were first used to transmit telegraph messages, such signals are also called amplitude (AT), frequency (FT), and phase (FT) telegraphy signals. Below, when describing the corresponding signals, this abbreviation will be used to distinguish them from signals with continuous modulation.

4.5.1. Discrete Amplitude Modulated Signals

Discrete amplitude modulation signals are characterized in that the amplitude of the carrier waveform changes in accordance with a control signal, which is a sequence of pulses, usually rectangular in shape. When studying the characteristics of signals with continuous modulation, a harmonic signal was considered as a control signal. By analogy with this, for signals with discrete modulation as a control signal, we use a periodic sequence of rectangular pulses

Obviously, as follows from (4.39), the pulse duration is, and the duty cycle.

In fig. 4.10 shows the diagrams of the control signal, the carrier oscillations and the amplitude-shift keyed signal. Hereinafter, we will assume that the amplitude of the pulses of the control signal is equal, and the initial phase of the carrier oscillation. Then the signal with discrete amplitude modulation can be written as follows

Earlier, the expansion of a sequence of rectangular pulses in the Fourier series (2.13) was obtained. For the case under consideration and expression (2.13) takes the form

Substituting (4.41) into (4.40) and using the formula for the product of cosines, we obtain:

In fig. 4.11 shows the amplitude spectrum of a signal amplitude modulated by a sequence of rectangular pulses. The spectrum contains a carrier frequency component with an amplitude and two sidebands, each of which consists of an infinite number of harmonic components located at frequencies whose amplitudes change according to the law ... The side bands, as in the case of continuous AM, are mirrored with respect to the spectral component of the carrier frequency. The zeros of the amplitude spectrum of the AT signal correspond to the zeros of the amplitude spectrum of the signal, but are shifted to the left and right by.

Due to the fact that the main part of the energy of the control signal is concentrated within the first lobe of the spectrum, the practical width of the spectrum in the case under consideration, based on Fig. 4.11 can be defined as

. (4.43)

This result is consistent with the spectrum calculations given in [L.4], where it is shown that most of the power is concentrated in the side components with frequencies and.

4.5.2. Discrete frequency modulated signals

When analyzing signals with discrete angular modulation, it is convenient to use a periodic sequence of rectangular pulses of the “meander” form as a modulating signal. Then the control signal on the time interval takes on the value , and on the time interval - the value. Again, as in the analysis of AT signals, we will assume.

As follows from subsection 4.3.1 a signal with frequency modulation is described by expression (4.24). Then, taking into account the fact that on the interval the control signal, and on the interval the control signal, after performing the integration operation, we obtain the expression for the QT signal

Figure 4.12 shows the timing diagrams of the control signal, the carrier waveform, and the discrete frequency modulation signal.

On the other hand, the FT signal, as follows from Fig. 4.12, can be represented by the sum of two signals of discrete amplitude modulation and, the frequencies of the carrier oscillations of which are respectively equal

,

While amplitude modulation changes the signal envelope in the "vertical plane", frequency modulation(FM) occurs in the "horizontal plane" of the signal. The amplitude of the carrier is kept constant and the frequency changes in proportion to the amplitude of the modulating signal.

Frequency deviation

The maximum amount by which the carrier frequency rises or falls under the influence of the amplitude of the modulating signal is called frequency deviation... This value depends solely on the amplitude (peak value) of the modulating voltage. In satellite TV broadcasting, the signal radiated to the Earth has a nominal frequency deviation of about 16 MHz / V and a bandwidth occupied by the information about the transmitted image of about 27 MHz.

Modulation index

Modulation index (t) is the ratio of the frequency deviation fd to the highest modulating frequency fm:

m = fd / fm.

Unlike amplitude modulation with FM, there is no need to limit the maximum value of the modulation index to one.

Johnson Noises

Noise is any unwanted random electrical disturbance. It permeates everywhere and is a major problem in electronics development. Such noise occurs in normal electrical circuits (measure after finishing plastering), especially in circuits with a resistor, at any temperature above zero Kelvin (0 K). This tiny, but not always insignificant, thermal noise, called Johnson noise, is detected (and can be measured as EMF) at the output ends of the circuit. The reason for the noise is the chaotic vibrations of molecules inside the resistor case, which cannot be stopped. Although the expression below is not particularly important in this case, it is worth considering to discover the relationship between EMF noise and temperature.

Johnson Noise RMS value = (4k tBR) ^ 1/2, where

t- absolute temperature in Kelvin (room temperature is about 290 K);
To- Boltzmann's constant t 1.38 x 10 ~ 23;
R- resistor value in ohms;
V- the bandwidth of the device for measuring the EMF value.

Calculating noise from a one megohm resistor at room temperature results in a value of about 0.4 mV. It may seem small, but its relative value is more important than its absolute value. If the useful signal is of the same order of magnitude as this value (and it can be much less), then it will drown in noise. According to the expression under consideration, which, by the way, applies not only to materials of artificial origin, the noise depends on the temperature and frequency band of the device for measuring its value. Such a device is a TV broadcast receiving station. Sidebands when transmitting a high quality signal are characterized by a large width, therefore, the receiving equipment must also have a wide frequency band for processing the incoming information. Under these conditions, the ingress of noise at the input of the circuit can severely limit the reception quality.

Signal to noise ratio

Signal to noise ratio (S / N) is the ratio of the EMF level of the wanted signal to the EMF level of any existing noise, which should be as high as possible. If the value of this ratio drops to one or less, then it is practically useless to transmit the signal. (In some cases, a rather expensive method for a computer to recreate the "signal environment" can be used, but this is not acceptable for a national satellite TV broadcasting system.)

Comparison of FM and AM

There are two properties of AM, due to which its use in the past was quite popular:

• the demodulation circuit in the receiver, called the rectifier, is fairly simple. Only a diode is required to cut one half-wave from the total signal and a low-pass filter to remove the residual carrier frequency;
• the sidebands are relatively narrow so that signal transmission does not take up too much space in the frequency spectrum.

The most serious disadvantage of AM is noise (or at least most of it), which consists of amplitude changes. In other words, any existing EMF noise is located at the top of the signal envelope, as shown in the figure.

Noise on AM signals

Therefore, to reduce the noise level, it is necessary either to increase the signal-to-noise ratio by more careful design of the receiving devices, or to use more crude methods that degrade the signal quality, for example, limiting the bandwidth.

On the other hand, FM is often considered noise-free, which is really wrong. FM transmission is also subject to noise as well as AM transmission. However, due to the method by which information is superimposed on the carrier frequency, most of the noise can be eliminated by the receiver circuitry. Since the noise is located on the outside of the FM signal, you can cut off the edges of the top and bottom of the received signal without disturbing information that is likely to be within the signal rather than at the edges. This clipping process is called amplitude clipping.

The disadvantage of FM is the requirement for a wide bandwidth for signal transmission. In fact, FM transmission is only possible when the carrier frequency is relatively high. Since satellite broadcasting is carried out at frequencies well above 1 GHz, this disadvantage can be considered insignificant.

It cannot be denied that the circuitry required to extract information from the FM carrier is quite complex, to put it mildly. A circuit that performs this function is called an FM demodulator. There are various circuitry for demodulating FM signals such as discriminators, ratio detectors, and phase-locked loop (PLL) circuits.

Decibels

In decibels (dB), the ratio between two powers can be expressed in another, often more convenient way. Instead of the actual relationship, the base 10 logarithm of the relationship is used:

dB = 10 log P1 / P2.

The result will be positive if Pt is greater than P2, and negative if P (less than P2. To eliminate the problem associated with calculating negative logarithms, the larger of the two cardinalities is put in the numerator, and the sign is determined later in accordance with the rule above.

Example
If P1 = 1000, and P2 = 10, then dB = 10 log 1000/10 = 10 log 100 = +20 dB.
(If P1 = 10 and P2 = 1000, the absolute value in decibels will be the same, but write it as -20 dB.).

Using decibels instead of actual ratios has the following advantages:

• since human hearing responds logarithmically to changes in sound intensity, the use of decibels is more natural. For example, if the output power of an audio amplifier increases from 10 to 100 W, this will not sound like a tenfold increase to the ear;
• decibels are useful for downsizing in large numbers. For example, a gain of 10,000,000 times would be only 70 dB;
• when passing from the antenna through the various stages in the receiver, the signal is amplified and lost. By expressing each gain and loss in positive and negative decibels, respectively, the overall gain can be easily calculated using algebraic addition. For example, (+5) + (-2) + (+3) + (-0.5) = 5.5 dB.

Below are some of the most commonly used decibel values.

We begin our analysis of the characteristics of angularly modulated signals by looking at single-tone frequency modulation. The control signal in this case is an oscillation of a unit amplitude (it can always be reduced to this form)

, (4.29)

and the modulated parameter of the carrier wave is the instantaneous frequency. Then, substituting (4.29) into (4.24), we obtain:

After performing the integration operation, we arrive at the following expression for the single-tone frequency modulation signal

Attitude

called index frequency modulation and has the physical meaning of the part of the frequency deviation per unit frequency of the modulating signal. So for example, if the deviation of the carrier frequency in MHz is , and the frequency of the control signal is kHz, then the frequency modulation index will be. In expression (4.30), the initial phase is not taken into account as having no fundamental significance.

The signal timing diagram for single-tone FM is shown in Fig. 4.7

We will start considering the spectral characteristics of the FM signal with a special case small frequency modulation index. Using the ratio

we represent (4.30) in the form

Since, then you can use the approximate representations

and expression (4.31) takes the form

Using the well-known trigonometric relation

and putting and, we get:

This expression resembles expression (4.6) for a single-tone AM signal. The difference is that if in a single-tone AM signal the initial phases of the side components are are the same, then in a single-tone FM signal at low frequency modulation indices they differ by an angle, i.e. are in antiphase.

The spectral diagram of such a signal is shown in Fig. 4.8

The values ​​of the initial phase of the lateral components are indicated in brackets. Obviously, the width of the spectrum of the FM signal at small indices of frequency modulation is equal to

.

Signals with low frequency modulation are rarely used in practical radio engineering.

In real radio engineering systems, the frequency modulation index significantly exceeds unity.

For example, in modern analog mobile communication systems that use frequency modulation signals for the transmission of voice messages at the upper frequency of the speech signal in kHz and frequency deviation kHz, the index, as it is easy to see, reaches a value of ~ 3-4. In VHF broadcasting systems, the frequency modulation index can exceed a value equal to 10. Therefore, let us consider the spectral characteristics of FM signals at arbitrary values ​​of the value.

Let's return to expression (4.32). The following types of decomposition are known

where is the Bessel function of the first kind of the th order.

Substituting these expressions in (4.32), after simple but rather cumbersome transformations using the already repeatedly mentioned relations of the products of cosines and sines, we obtain

(4.36)

where .

The resulting expression is the decomposition of a single-tone FM signal into harmonic components, i.e. amplitude spectrum. The first term of this expression is the spectral component of the carrier frequency oscillation with an amplitude ... The first sum of expression (4.35) characterizes the side components with amplitudes and frequencies, i.e. the lower sideband, and the second sum is the sidebands with amplitudes and frequencies, i.e. the upper side band of the spectrum.

The spectral diagram of the FM signal for an arbitrary one is shown in Fig. 4.9.

Let us analyze the nature of the amplitude spectrum of the FM signal. First of all, we note that the spectrum is symmetric with respect to the frequency of the carrier vibration and is theoretically infinite.

The components of the side side stripes are located at a distance Ω from each other, and their amplitudes depend on the frequency modulation index. And finally, for the spectral components of the lower and upper side frequencies with even indices, the initial phases coincide, and for the spectral components with odd indices they differ by an angle.

Table 4.1 shows the values ​​of the Bessel function for various i and . Let's pay attention to the component of the carrier vibration. The amplitude of this component is ... From table 4.1 it follows that at the amplitude, i.e. the spectral component of the carrier vibration in the spectrum of the FM signal is absent. But this does not mean that there is no carrier oscillation in the FM signal (4.30). The energy of the carrier vibration is simply redistributed between the components of the side bands.

Table 4.1

As already emphasized above, the FM spectrum - the signal is theoretically infinite. In practice, the bandwidth of radio devices is always limited. Let us estimate the practical width of the spectrum at which the reproduction of the FM signal can be considered undistorted.

The average power of the FM signal is defined as the sum of the average powers of the spectral components

The calculations showed that about 99% of the energy of the FM signal is concentrated in frequency components with numbers. This means that the frequency components with numbers can be neglected. Then the practical width of the spectrum for a single-tone FM, taking into account its symmetry with respect to

and for large values

Those. equal to twice the frequency deviation.

Thus, the width of the spectrum of the FM signal is approximately in times greater than the width of the spectrum of the AM signal. At the same time, for the transmission of information, it is used all energy signal. This is the advantage of FM signals over AM signals.

Balanced and SSB modulation

To more efficiently use the power of the AM signal spectrum, it is possible to exclude the carrier wave from the AM signal spectrum. Such an AM signal is called balance-modulated (BM). It is also possible to exclude one sideband (upper or lower) from the spectrum, since each of them contains complete information about the baseband signal. single sideband modulation - SSB.

FREQUENCY MODULATION

Angle modulation

The effect of the modulating signal on the argument (current phase) of the harmonic carrier is called angular modulation(MIND). Varieties of PA are frequency and phase.

19.2 Frequency modulation

Frequency modulation (FM) is the process of controlling the frequency of a harmonic carrier according to the law of a modulating signal.

The angular frequency changes according to the law:

where is the carrier frequency;

Deviation of the frequency of the modulated signal from the value;

Modulating signal. Can be harmonic (used for educational or research purposes) and non-harmonic (real signal);

Dimensional coefficient of proportionality, rad / (s ∙ V) or rad / (s ∙ A). Determined by the modulator circuitry.

The total phase at time t is found by integrating the frequency:

where is the phase incursion for the time from the origin to the considered moment;

Integration constant.

Mathematical model of the FM signal:

FM is called an integral type of modulation, because enters this expression under the integral sign.

Figure 19.1 - Timing diagrams of modulating, carrying and

modulated oscillation.

Harmonic FM

Consider harmonic FM(the modulating signal is harmonic).

The frequency changes according to the law:

where - frequency deviation at FM. Frequency deviation - the largest deviation of the frequency of the modulated signal from the value of the carrier frequency. With FM, it can take values ​​from units of hertz to hundreds of megahertz.

Phase at a point in time:

where is the frequency modulation index. It is a phase deviation at FM. Phase deviation is the greatest deviation of the phase of the modulated signal from the linear one.

Mathematical model of the signal at harmonic FM:

Using the trigonometric formula:, - we transform the expression:

Let us analyze separately for small and large modulation indices.

In the first case (), there are approximate equalities:

Using the trigonometric formula:, -

we arrive at the following expression for the FM signal:

Figure 19.2 - Spectral diagram of the FM signal at M FM<1.

With a small modulation index - narrowband FM- the amplitude spectral diagram of the FM signal coincides in composition (contains the central component with the carrier frequency, the lower and upper side components with the frequencies and) and the bandwidth () with the AM signal. The difference lies in the phase spectral diagram: the phase of the lower lateral component is shifted by 180 0.

With a small value of the modulation index, the advantages of FM (high noise immunity) will not appear. The spectrum width is the same as for AM.

In the second case () complex periodic functions: and - can be expanded in a Fourier series, and the FM signal can be represented as a sum of harmonic oscillations:

where is the Bessel function of the 1st kind of the nth order of the real argument. Tabulated;

n is the number of the harmonic component: the central component has n = 0, the side components - n = 1, 2, 3,….

Figure 19.3 - Spectrum of the FM signal at M FM = 2.

With a large modulation index - broadband FM- the spectrum of the FM signal consists of an infinite number of harmonics: from the component with the carrier frequency, the lower and upper side bands of frequencies formed by groups of components with frequencies and. In practice, only those lateral components are taken into account, the amplitudes of which are not less than 5% of the carrier amplitude, i.e. for which ... Then the width of the spectrum of the FM signal:.

This case is of main practical interest, since at high modulation indices, the noise immunity of signal transmission is significantly higher than with AM. The spectrum width of the FM signal is also significantly wider than that of AM.

With a complex baseband signal, the spectrum of the modulated signal turns out to be complex, containing different combination frequencies. The total frequency band occupied by such a signal:, where is the maximum frequency of the spectrum of the modulating signal; is the modulation index at this frequency.

PHASE MODULATION

Phase modulation

Phase modulation(FM) - change in the phase of the harmonic carrier according to the law of the modulating signal.

The instantaneous phase of the PM signal is determined by the expression:

where - deviation (shift) of the phase of the modulated signal from the linearly varying phase of the harmonic carrier;

Dimensional coefficient of proportionality, rad / V or rad / A.

Mathematical model of the FM signal:

The angular frequency is the rate of change (i.e., the time derivative) of the complete phase of the oscillation. Expression for the instantaneous frequency:

Thus, a baseband FM signal can be viewed as a baseband FM signal.

Figure 20.1 - Modulating signal, carrier oscillation, change in the phase of the PM signal, change in the frequency of the PM signal and the PM signal.

Harmonic FM

Consider the case of a harmonic baseband signal:

Phase of a harmonic PM signal:

where is the phase modulation index or phase deviation at FM. It can take a value from one to tens of thousands of radians.

Mathematical model of a signal with harmonic PM:

FM signal frequency:

where - frequency deviation at FM.

The calculation methodology and spectrum structure of the PM signal are similar to the FM signal, but the FM index must be replaced by the PM index. A similar close relationship between the spectra of FM and FM signals takes place in the case of nonharmonic modulating signals.

FM is used in the schemes of the indirect method of obtaining FM.

MANIPULATION

Types of manipulation

discrete modulation (keying)- modulation of a harmonic carrier wave with a discrete (digital) modulating signal. In this case, the modulated (informational) parameters of the carrier change abruptly. A device that implements the manipulation process is called a manipulator.

The discrete baseband signal is the primary signal representing the symbols of the code combinations of discrete messages. Examples of discrete primary signals: telegraph, data transmission, PCM.

There are the following types of manipulation:

Depending on the variable parameters of the carrier:

Amplitude (АМн; English term - amplitude shift keying, ASK),

Frequency (FMn; English term - frequency shift keying, FSK),

Phase (FMn; English term - phase shift keying, PSK),

Amplitude-phase keying (APSK; English term - APK / PSK, or amplitude phase keying, APK).

With AMn, each possible value of the transmitted symbol is associated with its own amplitude of the harmonic carrier oscillation, with FMn - frequency, with FMn - phase, and with AFMn - a combination of amplitude and initial phase;

Depending on the codes used:

Multi-position or -ary (in English - m-ary),

Binary (in English - binary).

Multiple Keying is used to increase the bit rate at the same modulation rate. - the basis of a multi-position code is the number of its different symbols. In practice, it is usually a nonzero power of two:, where is the number of binary digits (bits) representing the multiposition code symbols. Binary manipulation (,) is a special case of multipositional. As a rule, binary codes are used in discrete messaging systems.

Binary AMn

With a binary code, the primary signal takes two values, corresponding to code symbols 0 and 1:

- (-U m and, U m and) - bipolar signal;

- (0, U m and) - unipolar signal.

With binary AMn (BASK), symbol 1 corresponds to a segment of a harmonic carrier wave (message), symbol 0 - no oscillation (pause), therefore, AMn is often called passive pause manipulation.

Let's take a square wave signal as a modulating signal - a signal representing a sequence of bits of a repeating binary code 1010.

Figure 21.1 - Timing diagrams of the modulating and AMn signals.

AMn can be viewed as modulation by a signal with a spectrum rich in harmonics: the spectrum of a meander signal contains an infinite number of odd harmonics. The spectrum of the AMn signal contains a component with the carrier frequency and two sidebands, each of which repeats the spectrum of the primary signal.

Figure 21.2 - Spectral diagrams of modulating and AMn signals.

Theoretically, the signal spectrum for AMn is infinite. In practice, the infinite spectrum is limited by the filter bandwidth. The ratio for calculating the spectrum width of the AMn signal:

where is the symbol rate or modulation rate, Baud is the number of code symbols transmitted per unit of time (1 s);

Character (clock) interval - the time interval allotted for the transmission of one character.

AMn was invented in the early 20th century for wireless telegraphy. At present, AMn is no longer used in digital communication systems.

Binary FSK

With binary FSK (BFSK), symbol 1 corresponds to a segment of a harmonic oscillation with a frequency, and symbol 0 - with a frequency, where is the frequency deviation - the change in frequency during transmission 1 (0) relative to its average value. There is no passive pause with FMN, for this reason it is called active pause manipulation.

There are two possible FSK cases: continuous-phase FSK and CPFSK.

With phase-discontinuous FSK, the assignment of each binary symbol to its own frequency is arbitrary. The received signal contains phase jumps.

t

t

Figure 21.3 - Time signals: modulating and FSK with a phase gap.

The presence of phase discontinuities leads to "blurring" of the signal spectrum. This reduces the immunity of reception and interferes with other communication systems. Therefore, when choosing frequencies, it is necessary to ensure the condition of a smooth (without a phase jump) transition from a signal with a frequency to a signal with a frequency:

With binary PSK (BPSK), transmission 1 corresponds to a section of harmonic oscillation in phase with the carrier, and transmission 0 corresponds to 180 ° out of phase, i.e. the phase changes by 180 ° with each transition from 1 to 0 and vice versa.

t

Figure 21.6 - Timing diagram of the modulating and PSK signals.

PSK signal can be represented as the sum of two AMn signals, to obtain the first of which the carrier is used, and the second - ... The spectrum of the PSK signal amplitudes contains the same components as the AMn signal spectrum, except for the component with the carrier frequency (it disappears when the symbols 1 and 0 appear with equal probability). The amplitudes of the lateral components are approximately twice as large. When transmitting real codewords, the amplitude of the component with the carrier frequency is not zero, but will be significantly attenuated.

Figure 21.6 - Spectrum of PSK signal.

In DPSK, symbol 0 is transmitted by a segment of harmonic oscillation with the initial phase of the previous signal element, and symbol 1 is transmitted by the same segment with an initial phase that differs from the initial phase of the previous element by 180 ° (the phase changes when symbols 1 are transmitted), or vice versa (the phase changes when transfer of characters 0). In DPSK, transmission begins with the sending of one non-carrying element, which serves as a reference signal for comparing the phase of the next element.

Figure 21.7 - Timing diagram of the modulating and OFMn signal.

The spectrum of the PSK signal is similar to the spectrum of the PSK signal.

The PSK signal has the same bandwidth as the AMn signal:

.

FMN was developed at the beginning of the development of the deep space exploration program and is now widely used in commercial and military communications systems.

To the frequency of the modulating signal

Used in the document:

GOST 24375-80

Telecommunication vocabulary. 2013 .

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